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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Symplectic geometry\\[15mm]
\small lecture 5: Arnold Conjecture}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf HSE, room 306, 16:20,
\\[2mm] 
September 18, 2021
}
\end{center}

\newpage

{\bf \blue Lagrangian submanifolds (reminder)}

\definition
Let $X\subset M$ be a submanifold in a symplectic manifold $(M,\omega)$.
It is called {\bf \blue Lagrangian} if $T_x X \subset T_xM$
is Lagrangian for all $x\in X$. That is to say,
$\dim X = 1/2 \dim M$ and $\omega\restrict X=0$.

\theorem
Let $\xi\in \Lambda^1 M$ be a 1-form,
and $\Gamma_\xi\subset T^* M$ its graph, considered
as a submanifold in the total space of the cotangent
bundle. {\bf \red Then $\Gamma_\xi$ is Lagrangian
if and only if $d\xi=0$.}

\proof
Let $\sigma:\; x \mapsto (x, \xi(x))$ be the standard
diffeomorphism from $M$ to $\Gamma_\xi$.
Consider the restriction of $\theta$ to $\Gamma_\xi$.
For each  $u\in T_{(x, \xi(x))}\Gamma_\xi$,
the form $\theta$ takes $u$ to $\xi(d\pi(u))$.
This implies that $\sigma^* \theta\restrict{\Gamma_\xi} = \xi$,
hence $\sigma^* \omega\restrict{\Gamma_\xi} = d\xi$.
\endproof

\newpage

{\bf \blue Hamiltonian vector fields (reminder)}

\definition
Let $v\in TM$ be a vector field on a symplectic
manifold $(M, \omega)$. We say that
$v$ is {\bf \blue symplectomorphic} if $\Lie_v\omega=0$,
that is, if $\omega$ is invariant under the corresponding
diffeomorphism flow.

\remark From Cartan's formula, we have
$\Lie_v\omega=d (i_v\omega)$,
hence {\bf \purple $v$ is symplectomorphic if and only
if the $\omega$-dual 1-form is closed.}

\definition
Let $v\in TM$ be a symplectomorphic 
vector field on a symplectic manifold $(M, \omega)$,
and $\eta:= i_v\omega$ the corresponding 1-form.
We say that $v$ is {\bf \blue a Hamiltonian vector field}
if $i_v\omega$ is exact. Its {\bf\blue Hamiltonian}
is a function $f$ such that $df= i_v\omega$.
The group of {\bf \blue Hamiltonian symplectomorphisms} is
generated by diffeomorphisms obtained by exponents
of a time-dependent vector field $v_t$, which is Hamiltonian for all
$t\in [0,1]$.

\remark We have an exact sequence 
\[
0 \arrow \R \arrow C^\infty(M) \stackrel \delta \arrow
\Ham(M) \arrow 0
\]
If we identify  $\Ham(M)$ with exact 1-forms,
the differential $\delta:\; C^\infty(M)\arrow \Ham(M)$
{\bf \purple is identified with the de Rham differential.}

\newpage

{\bf \blue Hamiltonian vector fields on Lagrangian fibrations (reminder)}

\definition
Let $(M, \omega)$ be a symplectic manifold,
and $\pi:\; M \arrow X$ a smooth submersion.
It is called {\bf \blue a Lagrangian fibration}
if all its fibers are Lagrangian.

\claim Let $\pi:\; M \arrow X$ 
be a Lagrangian fibration, and $H$ a function on $X$.
Then {\bf \red the corresponding Hamiltonian vector field $v$
is tangent to the fibers of $\pi$.} Moreover, 
{\bf \red $v$ is non-degenerate
everywhere on a fiber $\pi^{-1}(x)$ 
if and only if $dH\neq 0$ in $x$;}
otherwise $v\restrict \pi^{-1}(x)=0$.

\proof Let $L:= \pi^{-1}(x)$.
Consider $\omega$ as a map from $T_m M$
to $T_m^* M$. Then $\omega^{-1}$ takes 1-forms vanishing
in $T_m L$ to vectors $v\in T_m M$ such that
$\omega(v, \cdot)$ vanishes on $T_mL$.
However, $T_mL^\bot= T_mL$ because $L$
is Lagrangian. {\bf \purple Therefore, $\omega^{-1}$
takes 1-forms vanishing on $T L$
to the vector fields tangent to $L$.}

The forms vanishing on $TL$ are generated by $\pi^* \Lambda^1 X$,
hence the corresponding Hamiltonian vector fields are tangent to
the fibers of $\pi$. 
\endproof


\remark 
Let $L= \pi^{-1}(x)$ be a fiber of a 
Lagrangian fibration. Then $\omega$ defines
a non-degenerate pairing between $T_mL$
and $T_x X$. This implies that {\bf\red the bundle
$TL$ is trivial.}


\newpage

{\bf \blue Hamiltonian isotopy of Lagrangian submanifolds}

Let $L_0, L_1$ be Lagrangian submanifolds in $(M, \omega)$.
We say that $L_0$ and $L_1$ are {\bf \blue Hamiltonian isotopic} if
there exists a flow $\Psi_t$ of Hamiltonian symplectomorphisms,
$\Psi_0 = \Id$, $t\in [0,1]$, such that $\Psi_1$ maps
$L_1$ to $L_0$.

\example
Let $\omega$ be the standard symplectic
form on the total space $T^*M$, and 
 $\pi:\; T^*M \arrow M$ the corresponding
Lagrangian projection. Consider an exact
1-form $\eta$ on $M$, and let $L_\eta\subset T^*M$
be its graph, considered as a Lagrangian submanifold.
Let $H\in C^\infty M$, with $\eta=dH$.
{\bf \purple The Hamiltonian vector field $v$ associated
with $H$ is tangent to fibers of $\pi$
and acts as a translation along each fiber.}
Evaluating $v$ at a fiber $\pi^{-1}(x)=T^*_x M$,
we obtain that $v\restrict x = \eta \restrict x$,
because $\omega(v, \cdot)=\eta$.
Then $e^v= \eta$, hence {\bf \purple the Hamiltonian
flow associated with $H$ takes the zero section
of $T^*M$ to $L_\eta$. }



\newpage

{\bf \blue Darboux' theorem}

\theorem
{\bf \red A symplectic manifold is locally symplectomorphic to a symplectic ball}
(in a neighbourhood of each point).

\pstep It is sufficient to check that for any
symplectic form $\omega_1$ on $\R^n$ there exists a neighbourhood 
$U\ni 0$ such that $(U, \omega_1)$ is symplectomorphic to 
a symplectic ball.

{\bf \green Step 2:} Choose coordinates  $x_i, y_i$ on $\R^{2n}$ 
in such a way that $\omega_1\restrict{T_0\R^{2n}}=\omega_0\restrict{T_0\R^{2n}}$,
where $\omega_0=\sum_i dx_i \wedge dy_i$. 
The form $\omega_t:=t\omega_1+(1-t)\omega_0$
is non-degenerate in 0, because $\omega_1 \restrict 0=\omega_0\restrict 0$. 
{\bf \purple Choose a starlike neighborhood $U\ni 0$ such that $\omega_t$
is non-degenerate for all $t\in [0,1]$.}

{\bf \green Step 3:} In $U$ the forms $\omega_t$ are all
non-degenerate and cohomologous. As in the proof of Moser's lemma, {\bf \purple choose
$\eta_t$ such that $\frac {d\omega_t}{dt}=d \eta_t$, 
and a vector field $v_t:=-\omega_t^{-1}(\eta_t)$,
vanishing in 0.}

Substracting from $\eta_t$ a constant 1-form, we
may assume that $\eta_t \restrict {T_0U}=0$. Then the
the coefficients of the form $\eta_t$
grow as $o(r)$, where $r$ is the distance from zero.
Therefore, for $U$ sufficiently small, 
{\bf \purple the vector field
$\Psi_t$ integrates in the whole $U$, and defines a
diffeomorphism $\Psi$ between $(U, \omega_0)$ and
$(\Psi(U), \omega_1)$.} Finally, since $v_t=0$ in 0,
the set $\Psi(U)$ contains 0.
\endproof



\newpage

{\bf \blue Weinstein neighbourhood theorem}

The following result is proven in the same way as the Darboux' theorem.

\theorem 
Let $X\subset M$ be a compact Lagrangian submanifold 
in $(M, \omega)$. 
Then {\bf \red there exists a neighbourhood $U$ of $X\subset M$
which is symplectomorphic to a neighbourhood
of $X$ in $X\subset T^* X$.}

\pstep 
Consider a smooth retraction (say, orthogonal projection)
$\pi:\; U\arrow X$. 
Since $X$ is Lagrangian, $\omega$ induces a non-degenerate
pairing between $TX$ and the fiberwise tangent bundle $T_\pi U$.
This gives a natural isomorphism
$T_\pi U\restrict X\cong T^* X$.
Using the fiberwise exponent map, we obtain
a diffeomorphism $\Psi$ between $U$ and a neighbourhood
of zero in $T_\pi U\restrict X= T^*X$.
{\bf \purple This diffeomorphism is compatible
with the symplectic structure on $TU\restrict X$.}

{\bf \green Step 2:} Let $\omega_0$ be the
symplectic structure on $U$ induced from
the embedding $U \arrow T^*X$, and $\omega_1$
the symplectic structure induced from $M$.
Consider the form $\omega_t:=t\omega_1+(1-t)\omega_0$, 
where $t\in [0,1]$. Since 
$\omega_0\restrict{TU\restrict X}=\omega_1\restrict{TU\restrict X}$,
{\bf \purple in a sufficiently small neighbourhood of $X$ all
$\omega_t$ are non-degenerate.} Shrinking $U$ if necessarily,
we may assume that this is true on all $U$.

\newpage

{\bf \blue Weinstein neighbourhood theorem (2)}


\theorem 
Let $X\subset M$ be a compact Lagrangian submanifold 
in $(M, \omega)$. 
Then {\bf \red there exists a neighbourhood $U$ of $X\subset M$
which is symplectomorphic to a neighbourhood
of $X$ in $X\subset T^* X$.}



{\bf \green Step 3:} Since $U$ is diffeomorphic to $X$,
and $X$ is Lagrangian, all $\omega_t$ are exact. Therefore 
we may choose a smooth family $\eta_t\in \Lambda^1 U$
of 1-forms such that $d\eta_t=\omega_t$. 
Denote by $j:\; X \arrow U$ the tautological embedding.
Since $j^*(\omega_t)=0$, one has $d(j^*\eta_t)=0$.
Replacing $\eta_t$ by $\pi^*(\text{closed 1-form})$
if necessarily, we may assume that $j^*\eta_t$ is exact.
Let $f_t\in C^\infty X$ be a family of functions
which satisfy $df_t =j^*\eta_t$. Replacing
$\eta_t$ by $\eta_t - d(\pi^*f_t)$, {\bf \purple we obtain
a family of 1-forms $\eta_t$ which satisfy
$d\eta_t = \omega_t$ and $j^*\eta_t=0$.}

{\bf \green Step 4:} Let $v_t:= \omega_t^{-1}(\eta_t)$,
and let $\Psi_t$ be the corresponding diffeomorphism flow.
Using Moser isotopy argument, we obtain that 
$\Psi_t^*\omega_0=\omega_t$. However,
 $\omega_t^{-1}$ maps the kernel of the
restiction map $\Lambda^1(U)\restrict X\arrow \Lambda^1X$
to $TX\subset TU\restrict X$, because $X$ is Lagrangian.
Therefore, $v_t$ preserves $X_t$. Putting a metric on $U$,
we may decompose the tangent bundle as $TU=T_\pi U\oplus T_\pi U^\bot$.
Let  $v_t^\pi$ be the part of $v_t$ which lies in $T_\pi U$.
Since $v_t^\pi$  vanishes in $X$, we have $|v_t^\pi|=o(r)$,
where $r$ is the distance to $X$. Using this estimate,
it is easy to see that {\bf \purple $v_t$ integrates to
a diffeomorphism flow in a sufficiently small neighbourhood of $X$
and maps it to another neighbourhood of $X$.}
\endproof


\newpage

{\bf \blue The flux of isotopic Lagrangian submanifolds}

\definition
Let $X\subset M$ be a compact Lagrangian submanifold,
and $U\supset X$ its Weinstein neighbourhood, 
identified with an open subset of $T^*X$.
We say that a sequence  $X_i\subset U$ of 
Lagrangian submanifolds {\bf \blue converges to $X$
in $C^\infty$-topology} if each $X_i$ is given 
as a graph of a closed 1-form $\eta_i$, and
$\eta_i$ converge to zero with all derivatives.

\definition
Define {\bf \blue the flux} on
the space of Lagrangian submanifolds
with $C^\infty$-topology as follows. Consider two isotopic
(bot not necessarily Hamiltonian isotopic)
submanifolds $L_0, L_1$, a homology
class $[u]\in H_1(M,\Z)$, and
homotopic circles $u_0\subset L_0$ an
$u_1\subset L_1$ representing $u$,
denote by $\psi:\; S^1\times [0,1]\arrow M$
the homotopy map. 
Then $\Flux_u(L_0, L_1):= \int_{S^1\times [0,1]}\psi^*\omega$.
Let $\Flux(L_0, L_1)$ be the associated map from
$H_1(M, \Z)$ to $\R$.


\newpage

{\bf \blue Flux: correctness of the definition}

\claim
Suppose that $\omega$ is exact. Then 
the map $\Flux_u(L_0, L_1)$ {\bf \purple is independent
from the choice of representatives $u_0, u_1$ and the 
homotopy $u_t$.}

\proof
Consider a class of the cohomology
of the pair $H^2(M, L_1\cup L_2)$
represented by $\omega$, and 
let $\delta:\; H^1(L_1\cup L_2) \arrow H^2(M, L_1\cup L_2)$
be the coboundary map of the exact sequence
\[
...\arrow H^1(L_1\cup L_2) \stackrel \delta 
\arrow H^2(M, L_1\cup L_2) \arrow H^2(M) \arrow H^2(L_1\cup L_2) \arrow ...
\]
Since $\omega$ is exact, 
$\omega=\delta(\rho)$, and flux is obtained 
as $\Flux_u(L_0, L_1)= \int_{u_0\cup -u_1}\rho$.
\endproof

\theorem
Let $L_0,L_1$ be Lagrangian submanifolds
in $(M,\omega)$. Suppose that $L_1$ is
a identified with a graph of a closed 1-form
$\eta$ in a Weinstein neighbourhood  $U\subset L_0$.
{\bf \red Then $L_1$ is Hamiltonian isotopic to $L_0$
if and only if $\Flux(L_0, L_1)$.}
Moreover, {\bf \red every element of $\Hom(H_1(M, \Z), \R)$
is represented by a flux of a Lagrangian submanifold.}

{\bf \green The proof follows from Claim 1 on the
next slide.}

\newpage

{\bf \blue Hamiltonian isotopy of Lagrangian submanifolds in $T^*X$}

{\bf \green Claim 1:}
Let $\eta$ be a closed 1-form on $M$,
and $L_\eta\subset T^*M$
be its graph, considered as a Lagrangian submanifold.
{\bf \red Then $L_\eta$ is Hamiltonian isotopic to the
zero section $L_0$ of $T^*M$ if and only if $\eta$ is exact.}

\proof
We have already obtained the Hamiltonian isotopy 
from exactness of $\eta$. To prove the converse
implication, we need to show that $L_\eta$
is not Hamiltonian isotopic to $L_0$ when $\eta$
is not exact. {\bf \purple To prove this we compute 
$\Flux(L_0,L_\eta)$ and show that it is
non-zero unless $\eta$ is exact. }

Let $\Psi_t$ map $(x, \xi)$
to $(x, \xi+ t\eta)$, and let $v$ be the vector
field tangent to this action, 
$v\restrict (x, \xi)=(0, \eta)$. This
vector field acts on $T^*M$ by symplectomorphisms
because it is locally in $M$ Hamiltonian.

{\bf \purple To finish the proof, it suffices to show that
 $\Flux_{[u]}(L_0,L_\eta)=\int_{[u]}\eta$ for any
 $[u]\in H^1(M,\Z)$. }

Let $u$ be its representative
in the zero section of $T^* M$, with
$u:\; S^1 \arrow M\subset T^* M$. The corresponding annulus
$A:=\Psi_{[0,1]}(u)$ has boundaries in $u$ and
$\Psi_1(u)$. Using $d\theta =\omega$, we obtain
$\int_A \omega= -\int_{u}\theta + \int_{\Psi_1(u)}\theta$.
The first integral vanishes, because $\theta=0$
on the zero section, and the second
integral is equal to $\int_u\eta\neq 0$ because
$\theta = \eta$ on $L_\eta$. 
We proved that $L_\eta$ is not Hamiltonian 
isotopic to the zero section.
\endproof



\newpage

{\bf \blue Arnold Conjecture}

\remark Let $\eta$ be an exact form, 
and $L_\eta\subset T^*M$ its graph, considered
as a Lagrangian submanifold in $T^*M$. Denote by $L_0$ the
zero section of $T^* M$. The Lagrangian submanifolds
$L_\eta$ and $L_0$ intersect transversally if 
and only if $\eta= dH$, where $H$ is a Morse function.
Clearly, {\bf \purple the number of intersection points $L_0 \cap L_\eta$
is equal to the number of critical points of a Morse
function, hence $\#(L_0 \cap L_\eta) \geq \sum_i b_i(M)$.}

\conjecture {\bf \blue (Arnol'd, proven by Floer)}\\
Let $L\subset T^*M$ be a Lagrangian subvariety
which transversally intersects the zero section
$L_0\subset T^* M$. Suppose that $L$ is Hamiltonian
isotopic to $L_0$. {\bf \red Then 
$\#(L_0 \cap L) \geq \sum_i b_i(M)$.}


\newpage

{\bf \blue Strong Arnold Conjecture}


\remark
Note that {\bf \purple the minimal number $m(M)$ of critical points
of Morse functions on $M$ is, generally speaking, strictly 
bigger than $\sum_i b_i(M)$.}

The strong Arnold conjecture is still open.

\conjecture {\bf \blue (strong Arnol'd conjecture)}\\
Let $L\subset T^*M$ be a Lagrangian subvariety
which transversally intersects the zero section
$L_0\subset T^* M$. Suppose that $L$ is Hamiltonian
isotopic to $L_0$. {\bf \red Then 
$\#(L_0 \cap L) \geq m(M)$.}

\conjecture {\bf \blue (Arnol'd-Givental conjecture)}\\
Let $(M, \omega)$ be a symplectic manifold,
and $L_1, L_2$ Hamiltonian isotopic Lagrangian submanifolds,
intersecting transversally. {\bf \red Then\\
$\#(L_1 \cap L_2) \geq \sum_i \dim H^i(M,\Z/2)$.}



\end{document}


